
theorem
  for X being finite set, F being Dependency-set of X, K being Subset of
X st F = { [A, B] where A, B is Subset of X : K c= A or B c= A } holds {X}\/{B
  where B is Subset of X : not K c= B} = saturated-subsets F
proof
  let X be finite set, F be Dependency-set of X, K being Subset of X;
  set BB = {X}\/{B where B is Subset of X : not K c= B};
  set M = {[K, X]}\/{[A, A] where A is Subset of X : not K c= A};
  set ss = saturated-subsets F;
  assume F = { [A, B] where A, B is Subset of X : K c= A or B c= A };
  then
A1: M = Maximal_wrt F by Th30;
A2: [#]X = X;
  now
    let x be object;
    hereby
      assume
A3:   x in BB;
      per cases by A3,XBOOLE_0:def 3;
      suppose
A4:     x in {X};
        [K,X] in {[K,X]} by TARSKI:def 1;
        then [K,X] in M by XBOOLE_0:def 3;
        then
A5:     K ^|^ X, F by A1;
        x = X by A4,TARSKI:def 1;
        hence x in ss by A2,A5;
      end;
      suppose
        x in {B where B is Subset of X : not K c= B};
        then consider B being Subset of X such that
A6:     x = B and
A7:     not K c= B;
        [B,B] in {[A, A] where A is Subset of X : not K c= A} by A7;
        then [B,B] in M by XBOOLE_0:def 3;
        then B ^|^ B, F by A1;
        hence x in ss by A6;
      end;
    end;
    assume x in ss;
    then consider b, a being Subset of X such that
A8: x = b and
A9: a ^|^ b, F by Th31;
A10: [a,b] in M by A1,A9;
    per cases by A10,XBOOLE_0:def 3;
    suppose
      [a,b] in {[K,X]};
      then [a,b] = [K,X] by TARSKI:def 1;
      then b = X by XTUPLE_0:1;
      then b in {X} by TARSKI:def 1;
      hence x in BB by A8,XBOOLE_0:def 3;
    end;
    suppose
      [a,b] in {[A, A] where A is Subset of X : not K c= A};
      then consider c being Subset of X such that
A11:  [a,b] = [c,c] and
A12:  not K c= c;
A13:  c in {B where B is Subset of X : not K c= B} by A12;
      b = c by A11,XTUPLE_0:1;
      hence x in BB by A8,A13,XBOOLE_0:def 3;
    end;
  end;
  hence thesis by TARSKI:2;
end;
